Control and observation of neutral systems.

*(English)*Zbl 0546.93041
Research Notes in Mathematics, 91, Boston-London-Melbourne: Pitman Advanced Publishing Program. 207 p. £9.95 (1984).

These research notes are devoted to mathematical studies of control and observation problems for dynamical systems described by linear functional differential equations of neutral type. The two basic mathematical models are
\[
(1)\quad d/dt(x(t)-Mx_ t-\Gamma u_ t)=Lx_ t+Bu_ t
\]
for control problems and
\[
(2)\quad d/dt(x(t)-M^ Tx_ t)=L^ Tx_ t,\quad y(t)=B^ Tx_ t+\Gamma^ T\dot x_ t
\]
for observation problems. Here all time dependent vectors are with values in finite dimensional real spaces, \(x_ t\) and \(u_ t\) are restrictions of the maps x(.) and u(.), resp., to the interval [t-h,t], h is the maximal delay present in the system, the operators M, \(\Gamma\), L, B are represented by Volterra-Stieltjes integrals with kernels of bounded variation, and the corresponding ”transposed” operators have transposed matrices as kernels. The main mathematical tool used in the book is the theory of one-parameter semigroups of linear operators representing the transition of an initial state of system (1) (or (2)) to a final state.

Despite the fact that a fairly extensive theory for general \(C_ 0\)- semigroups of linear operators exists [cf. E. Hille and R. S. Phillips, ”Functional analysis and semigroups” (1957; Zbl 0078.100)] not much is directly applicable to the field of delay-differential systems, since the fundamental problem in this field, if we assume the semigroup viewpoint, is a proper choice of a particular function space as a state space, and discovering and exploiting specific properties of systems with time delays. For examining fundamental problems of existence, uniqueness and continuous dependence of solutions to functional-differential equations (FDE) as well as their stability the most popular has been the choice of the space of continuous functions as the state space [cf. J. K. Hale, Theory of functional differential equations (1977; Zbl 0352.34001)]. However, for control and observation problems it appears that more useful is \(M^ p=R^ n\times L^ p\) where the state \(x_ t\) in \(M^ p\) is partitioned as \(x_ t=(x(t),x_ t)\) where \(x_ t\) is the restriction of x to [t-h,t). \(M^ p\), contrary to C, admits discontinuous functions and can be a candidate for a state space of quite general systems as, e.g., (1) (note that for \(L=0\), \(B=0\), \(\Gamma =0\) a typical trajectory of (1) is a discontinuous one). The author adopts the space \(M^ p\) and suggests a second space, namely the Sobolev space \(W^{1,p}\), as a state space for systems (1), (2). Working with two spaces appears to be particularly useful, especially in completing the duality theory which is one of the author’s original results.

The contents of the book is as follows. Chapter 1 (Preliminaries) is a short (34 pages) introduction to the theory of linear Volterra-Stieltjes integral equations, equations of form (1), and the basic properties of controlled and observed differential equations in reflexive Banach spaces. Basic properties of convolutions of functions of bounded variation and the corresponding Borel measures are recalled. Existence, uniqueness and continuous dependence of solutions on the inhomogeneous term are proved for integral equations. Similar results are proved for equation (1) and for two coupled equations: one functional and one FDE. A corresponding result on generation of a \(C_ 0\)-semigroup in the space \(M^ p\) is presented as well as elementary properties of the infinitesimal generator. Next, some basic results on semigroups generated by differential equations in reflexive Banach spaces are given, especially by equations with feedback control and by observer equations.

Chapter 2 (State space theory for neutral functional differential systems) is devoted to the detailed development of semigroup theory for equation (1) in spaces \(M^ p\) and \(W^{1,p}\). After introducing the semigroup for homogeneous versions of (1) and (2) the mutually dual state concepts are considered. Next structural operators are introduced to exhibit how the present state is generated by the immediate past trajectory values. Representation of solutions (fundamental solution) is discussed. It is shown how dual state concepts can be applied to controlled (1) and observed (2) equations. Structural operators related to output y and input u are introduced. Finally, the spectral properties of semigroups are recalled.

In Chapter 3 (Completeness and small solutions) the central problems are completeness of the set of eigenfunctions as a property of generating a dense linear subspace and the description of small solutions (vanishing in finite time). A duality theorem is proved: completeness for (1) means nonexistence of nonzero small solutions for (2). Also the modified version of this theorem is given where F-completeness (F \(=\) a structural operator) is examined.

Chapter 4 (Controllability and observability) is devoted to the study of relationships between various controllability and observability concepts. Spectral controllability is defined as controllability of all finite dimensional projections. Similarly, spectral observability is defined. Appropriate rank criteria in terms of the characteristic matrix are given. Next, duality between approximate controllability and strict observability is demonstrated. Observability of small solutions is examined (rank criteria). Completability as the possibility of getting a complete set of eigenfunctions after a feedback appears as an important condition in characterizing approximate controllability. Duality between approximate F-controllability and observability is shown.

In Chapter 5 (Feedback stabilization and dynamic observation) first the stabilization problem is examined in two steps: 1) stabilization of the difference part, 2) shifting the remaining finite number of eigenvalues. Step 1) is done effectively only for finitely many commensurable delays in the difference part; the generalization is still an open problem. Next, the perturbed semigroups arising in the state feedback and dynamic observation problems are examined and characterized. Also, it is proved that, for systems with stable difference part, the possibility of shifting anywhere any finite part of the spectrum by a proper choice of state feedback is equivalent to spectral controllability. A dynamic output feedback via an observer and a feedback from the estimate of the state is suggested.

The Appendix contains a criterion for observability of inputs (in finite dimensional systems) used formerly in the text. 152 references are given. Bibliographical notes are either at the beginning or the end of each chapter.

Despite the fact that a fairly extensive theory for general \(C_ 0\)- semigroups of linear operators exists [cf. E. Hille and R. S. Phillips, ”Functional analysis and semigroups” (1957; Zbl 0078.100)] not much is directly applicable to the field of delay-differential systems, since the fundamental problem in this field, if we assume the semigroup viewpoint, is a proper choice of a particular function space as a state space, and discovering and exploiting specific properties of systems with time delays. For examining fundamental problems of existence, uniqueness and continuous dependence of solutions to functional-differential equations (FDE) as well as their stability the most popular has been the choice of the space of continuous functions as the state space [cf. J. K. Hale, Theory of functional differential equations (1977; Zbl 0352.34001)]. However, for control and observation problems it appears that more useful is \(M^ p=R^ n\times L^ p\) where the state \(x_ t\) in \(M^ p\) is partitioned as \(x_ t=(x(t),x_ t)\) where \(x_ t\) is the restriction of x to [t-h,t). \(M^ p\), contrary to C, admits discontinuous functions and can be a candidate for a state space of quite general systems as, e.g., (1) (note that for \(L=0\), \(B=0\), \(\Gamma =0\) a typical trajectory of (1) is a discontinuous one). The author adopts the space \(M^ p\) and suggests a second space, namely the Sobolev space \(W^{1,p}\), as a state space for systems (1), (2). Working with two spaces appears to be particularly useful, especially in completing the duality theory which is one of the author’s original results.

The contents of the book is as follows. Chapter 1 (Preliminaries) is a short (34 pages) introduction to the theory of linear Volterra-Stieltjes integral equations, equations of form (1), and the basic properties of controlled and observed differential equations in reflexive Banach spaces. Basic properties of convolutions of functions of bounded variation and the corresponding Borel measures are recalled. Existence, uniqueness and continuous dependence of solutions on the inhomogeneous term are proved for integral equations. Similar results are proved for equation (1) and for two coupled equations: one functional and one FDE. A corresponding result on generation of a \(C_ 0\)-semigroup in the space \(M^ p\) is presented as well as elementary properties of the infinitesimal generator. Next, some basic results on semigroups generated by differential equations in reflexive Banach spaces are given, especially by equations with feedback control and by observer equations.

Chapter 2 (State space theory for neutral functional differential systems) is devoted to the detailed development of semigroup theory for equation (1) in spaces \(M^ p\) and \(W^{1,p}\). After introducing the semigroup for homogeneous versions of (1) and (2) the mutually dual state concepts are considered. Next structural operators are introduced to exhibit how the present state is generated by the immediate past trajectory values. Representation of solutions (fundamental solution) is discussed. It is shown how dual state concepts can be applied to controlled (1) and observed (2) equations. Structural operators related to output y and input u are introduced. Finally, the spectral properties of semigroups are recalled.

In Chapter 3 (Completeness and small solutions) the central problems are completeness of the set of eigenfunctions as a property of generating a dense linear subspace and the description of small solutions (vanishing in finite time). A duality theorem is proved: completeness for (1) means nonexistence of nonzero small solutions for (2). Also the modified version of this theorem is given where F-completeness (F \(=\) a structural operator) is examined.

Chapter 4 (Controllability and observability) is devoted to the study of relationships between various controllability and observability concepts. Spectral controllability is defined as controllability of all finite dimensional projections. Similarly, spectral observability is defined. Appropriate rank criteria in terms of the characteristic matrix are given. Next, duality between approximate controllability and strict observability is demonstrated. Observability of small solutions is examined (rank criteria). Completability as the possibility of getting a complete set of eigenfunctions after a feedback appears as an important condition in characterizing approximate controllability. Duality between approximate F-controllability and observability is shown.

In Chapter 5 (Feedback stabilization and dynamic observation) first the stabilization problem is examined in two steps: 1) stabilization of the difference part, 2) shifting the remaining finite number of eigenvalues. Step 1) is done effectively only for finitely many commensurable delays in the difference part; the generalization is still an open problem. Next, the perturbed semigroups arising in the state feedback and dynamic observation problems are examined and characterized. Also, it is proved that, for systems with stable difference part, the possibility of shifting anywhere any finite part of the spectrum by a proper choice of state feedback is equivalent to spectral controllability. A dynamic output feedback via an observer and a feedback from the estimate of the state is suggested.

The Appendix contains a criterion for observability of inputs (in finite dimensional systems) used formerly in the text. 152 references are given. Bibliographical notes are either at the beginning or the end of each chapter.

Reviewer: A.W.Olbrot

##### MSC:

93C25 | Control/observation systems in abstract spaces |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

34K35 | Control problems for functional-differential equations |

93B05 | Controllability |

93B07 | Observability |

47D03 | Groups and semigroups of linear operators |

46B10 | Duality and reflexivity in normed linear and Banach spaces |

93D15 | Stabilization of systems by feedback |

93C05 | Linear systems in control theory |

93B03 | Attainable sets, reachability |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |